Show that $a^4(b^2-c^2) + b^4(c^2-a^2)+c^4(a^2-b^2)$ is divisible by $(a+b)(b+c)(c+a)(a-b)(b-c)(c-a)$.

Question

Show that $$a^4(b^2-c^2) + b^4(c^2-a^2)+c^4(a^2-b^2)$$ is divisible by $$(a+b)(b+c)(c+a)(a-b)(b-c)(c-a).$$

Answer 1

Note that substituting any of the following yields 0: $a=b, b=c, c=a, a=-b, b=-c, c=-a$. Thus by the factor theorem (which still holds on multivariate polynomials), we have $a-b, b-c, c-a, a+b, b+c, c+a$ are factors of the expression respectively.

Answer 2

Your term is $$(a-b) (a+b) (a-c) (a+c) (b-c) (b+c)$$